If you like Rush, but like a bit of the heavier side of DT, I think I might recommend the following first:
-Scenes from a Memory: This album is almost always in most fans' #1 or #2 spot. It is a concept album, but also just has pretty strong individual songs. Some of the musical themes are intentionally derived from the song Metropolis, pt. I, which is from their Images & Words album, so many might recommend that as a starting point. The only reasons I am hesitant to recommend that is a first choice for you is, (1) the production sounds dated (it is from 1992), and (2) while it has their biggest hit (Pull Me Under) and plenty of other strong songs, it does have a couple of sappy ballads.
-Images & Words: See above. If the two issues I mentioned are not a problem for you, you will probably love this album.
-Awake: Fairly concise songs (for Dream Theater). This album is dark and moody. There is plenty of heaviness. The production of the album is a little different, and can catch a new listener off guard a bit at first. If you are a fan of classic Queensryche (which I am guessing you may be based on how long you have been a Rush fan), it's sort of similar to how "harsh" the production values sound on Mindcrime. It is jarring the first time or two that you listen, which can be distracting. But after a few listens, you generally come to understand how it fits the album, and it transforms into a positive rather than a potential negative.
-A Dramatic Turn of Events: Somewhat similar to the self-titled album you have, but in many ways, also calls back to Images & Words in a more modern way.
-Six Degrees of Inner Turbulence: FANTASTIC ALBUM. This one is my personal #1. It is a double album, but only six songs, as the last song is 43 minutes long (divided into several movements that are separated with "courtesy track breaks" so fans can go straight to certain movements without listening to the entire thing if they choose). Overall, the album is heavy. It is experimental. It represents the best of what makes Dream Theater who they are as a band, in my opinion. The only reason they are my #5 choice for you instead of #1 is because, as a new fan, I could see the idea of a 43 minute song to be offputting.
I will say that, pretty much no matter which album you go with, the things you did not like "the soft piano, the lyrics, the interludes, and the song lengths," will be present to varying degrees in all of them. The song lengths are something that will likely grow on you the more you become familiar with their music. They do heavy and complex very well. But they have a pretty wide variety styles and influences that they like to bring to their music, so there is generally always something to love and something to hate on every album.